# Finding Exact Value of Trig Expression

• whatisphysics
In summary, the conversation discusses finding the exact values of two expressions, sec-1(\sqrt{}2) and sin-1(1), and using inverse functions to solve for these values. The formula for sec\theta=\stackrel{}{}1/cos\theta is mentioned, and the concept of inverse functions is explained. The conversation ends with a suggestion to memorize the unit circle to assist with finding the values of trigonometric functions.
whatisphysics

## Homework Statement

Find the exact value of each expression:
a) sec-1($$\sqrt{}2$$)
b) sin-1(1)

## Homework Equations

sec$$\theta$$=$$\stackrel{}{}1/cos\theta$$

## The Attempt at a Solution

I've never learned this, but I am really curious in how it is solved.
Is there a formula for this? Thanks!

These are inverse functions so:

a. For what value of x does sec(x) = sqrt(2)

b. For what value of x does sin(x) = 1?

whatisphysics said:

## Homework Statement

Find the exact value of each expression:
a) sec-1($$\sqrt{}2$$)
b) sin-1(1)

## Homework Equations

sec$$\theta$$=$$\stackrel{}{}1/cos\theta$$

## The Attempt at a Solution

I've never learned this, but I am really curious in how it is solved.
Is there a formula for this? Thanks!
Do you understand inverse functions?

IOW, x = f-1(y) <==> y = f(x)

For example, suppose you were asked to find cos-1(1/2).

Let y = cos-1(1/2).
That is equivalent to 1/2 = cos(y). What angle in the interval [0, $\pi$] has a cosine of 1/2?

Mark44 said:
Do you understand inverse functions?

IOW, x = f-1(y) <==> y = f(x)

For example, suppose you were asked to find cos-1(1/2).

Let y = cos-1(1/2).
That is equivalent to 1/2 = cos(y). What angle in the interval [0, $\pi$] has a cosine of 1/2?

Should I memorize the circle with all the angles?
And this may sound silly...but on (x,y), which is cos and sin? Is it like (cos, sin) on the circle, or the opposite?

On the unit circle, x = cos(t) and y = sin(t).

A simple way to look at the problem is let $$a=\sec^{-1}\sqrt{2}$$ then $$\sec a=\sqrt{2}$$. From here it is easy to compute the value a by turning sec into cos and using information about known values of cos.

Last edited:
Thank you all for the input! I think I will learn to memorize the circle with all the angles...I'm sure that will help.

## What is the purpose of finding the exact value of a trigonometric expression?

The purpose of finding the exact value of a trigonometric expression is to simplify the expression and make it easier to work with in mathematical calculations. It also allows for a more precise answer instead of an approximate value.

## What are the basic trigonometric identities used to find exact values?

The basic trigonometric identities used to find exact values are sine, cosine, and tangent. These identities relate the sides and angles of a right triangle and can be used to find the exact values of trigonometric expressions.

## What are the steps to finding the exact value of a trigonometric expression?

The steps to finding the exact value of a trigonometric expression are:

• 1. Rewrite the expression using basic trigonometric identities.
• 2. Simplify the expression using algebraic rules and properties.
• 3. Substitute the given values for the variables in the expression.
• 4. Use a calculator or reference table to find the exact value of the trigonometric functions.
• 5. Simplify the final answer, if necessary.

## How do you know if you have found the exact value of a trigonometric expression?

You can determine if you have found the exact value of a trigonometric expression by checking if the answer is in the form of a fraction or a radical. If the answer is a decimal or an approximation, you have not found the exact value.

## Can the exact value of a trigonometric expression be irrational?

Yes, the exact value of a trigonometric expression can be irrational, meaning it cannot be written as a fraction of two integers. This is because some trigonometric functions, such as sine and cosine, have irrational values for certain angles, such as 30 degrees or 45 degrees.

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